Problem 18 - College Algebra

Question

For exercises 17 and 18, use this scenario: A retired woman has \(\$ 50,000\) to invest but needs to make \(\$ 6,000\) a year from the interest to meet certain living expenses. One bond investment pays 15\(\%\) annual interest. The rest of it she wants to put in a CD that pays 7\(\% .\) Set up and solve the equation for how much the woman should invest in each option to sustain a \(\$ 6,000\) annual return.

Step-by-Step Solution

Answer

Invest \$31,250 in the bond and \$18,750 in the CD.

Step 1: Define Variables

Let's define two variables: Let \( x \) be the amount invested in the bond paying 15% interest. Let \( y \) be the amount invested in the CD paying 7% interest.

Step 2: Set Up Equations

Using the given information, we have two conditions: 1. The total investment is \( x + y = 50,000 \).2. The total annual interest earned should be \( 0.15x + 0.07y = 6,000 \). These are the two equations we will solve.

Step 3: Express y in terms of x

From the first equation \( x + y = 50,000 \), express \( y \) in terms of \( x \): \[ y = 50,000 - x \]

Step 4: Substitute y in Interest Equation

Substitute \( y = 50,000 - x \) into the interest equation: \[ 0.15x + 0.07(50,000 - x) = 6,000 \] Simplify this to get an equation with just \( x \).

Step 5: Solve for x

Calculate:\[ 0.15x + 3,500 - 0.07x = 6,000 \] Combine like terms:\[ 0.08x = 2,500 \] Solve for \( x \): \[ x = \frac{2,500}{0.08} = 31,250 \] So, the amount to invest in the bond is \( 31,250 \) dollars.

Step 6: Solve for y

Now that we have \( x = 31,250 \), use this to find \( y \): \[ y = 50,000 - 31,250 = 18,750 \]Therefore, the amount to invest in the CD is \( 18,750 \) dollars.

Key Concepts

Algebraic Equations

Algebraic equations are the backbone of solving many problems, including investment problems like the one presented. In this problem, we use equations to represent the relationship between different financial actions and outcomes.

• The first equation represents the total investment sum: \( x + y = 50,000 \), where \( x \) and \( y \) are the amounts invested in bonds and CDs respectively.
• The second equation involves the interest income: \( 0.15x + 0.07y = 6,000 \), indicating the total interest needed.

To find how much should be invested where, we express one variable in terms of the other (like \( y = 50,000 - x \)) and substitute it back into the other equation. This simplifies the problem to one equation with one unknown, which is easier to solve. By solving these equations, different scenarios and conditions can be met using real numbers.

Interest Calculation

Interest calculation is essential for determining how much money an investment will earn over time. In this problem, we look at two kinds of investments, each providing a different interest rate:

• Bonds: These offer 15% interest per annum, making them more lucrative but possibly riskier.
• Certificates of Deposit (CDs): These offer 7% interest per annum, providing a more stable but lower return.

The interest for each investment is calculated using the formula: \( \,\text{Interest} = \text{Principal} \times \text{Rate}\,\). For example, if \( x \) dollars are invested in bonds, the interest is \( 0.15x \), whereas for the \( y \) dollars in CDs, it is \( 0.07y \). We've set these to total the necessary \( 6,000 \) annual income using the equation \( 0.15x + 0.07y = 6,000 \).

Investment Strategies

Choosing the right investment strategy is crucial for achieving financial goals. In our example, the retired woman needs to generate \( 6,000 \) dollars annually from a total \( 50,000 \) dollars investment.

• High-Interest Bonds: These are chosen to maximize returns at a 15% rate. Despite potential greater returns, they may carry higher risks.
• Low-Interest CDs: A safer option with a 7% yield, CDs offer stability and guarantee return, making them reliable for risk-averse strategies.

By investing \( 31,250 \) dollars in bonds and \( 18,750 \) dollars in CDs, as calculated, the woman balances between high returns and security. This mix capitalizes on the higher bond interest while securing part of her portfolio in safer CDs, ensuring a reliable annual income to meet expenses.